From Philosophical Logic to Computer Science Aldo Antonelli - - PowerPoint PPT Presentation

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

From Philosophical Logic to Computer Science — and back again

• G. Aldo Antonelli

aldo.antonelli@uci.edu

• Dept. of Logic & Philosophy of Science

University of California, Irvine

Logic Colloquium Wroclaw, July 14-19, 2007

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

Outline

Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

Philosophical Logic

◮ Philosophical Logic is not distinguished from

Mathematical Logic either in methods or in subject matter (it is, in each case, the study of formal languages by mathematical methods), but rather in inspiration.

◮ Traditionally, Philosophical Logic derived its problems

from the analysis of philosophical issues. In this, Philosophical Logic was part and parcel with the linguistic turn in philosophy — the idea that many traditional philosophical problems can be explained (or explained away) by linguistic analysis. Logic — this time simpliciter — provided the formal tools for philosophical analysis.

◮ This enterprise was not initially regarded as

conceptually distinct from the application of logical tools to the analysis of mathematical reasoning. This unity was reflected in 1936 when the ASL was founded — it was all, perhaps redundantly, symbolic logic.

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

The logic of possibility and necessity

◮ Modal Logic has quintessentially philosophical origins

in the study of the alethic modalities: possibility and necessity.

◮ Philosophers have dealt with modalities ever since

Aristotle, but especially with Leibniz and Kant (both

• f whom recognized the duality of possibility and

necessity).

◮ Modal logic began with Lewis & Langford’s Symbolic

Logic (!) (1932). L&L argued against Russell’s use of the material conditional A ⊃ B in Principia in favor of necessary implication ✷(A ⊃ B).

◮ This has led to the development of the philosopher’s

favorite style (“plain vanilla”) of (mono-) modal logic and its different system, K, T, B, S4, S5, . . .

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

Modal Logics

◮ Just like there is more to quantifiers than ∃ and ∀, so

there is more to modal logic than ✷ and ✸.

◮ One useful characterization of modal logic is that is

perhaps the simplest way to describe relational structures, i.e., structures of the form (A, R), where R ⊆ A2.

◮ Of course there are many ways to talk about

relational structures, beginning with first- and second-order logic. Modal logic differs from all these by taking an internal viewpoint, i.e., by asking what the structure looks like from within.

◮ The difference is that not all of the structure (A, R)

may be accessible from any given point a ∈ A. This expressive limitation has proved immensely fruitful.

◮ A further characterization is due to Tarski, who

analyzed modal logic in terms of Boolean algebras with operators.

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

Transition systems

◮ Perhaps the deepest and broadest characterization

regards modal logic as the theory of various kinds of transitions (represented by ✷) between states of a given system.

◮ ✷A is true at state s iff A is true at every state s′ ← s.

(Dually for ✸A).

◮ As an immediate consequence, the schema K is valid:

✷(A ⊃ B) ⊃ (✷A ⊃ ✷B).

◮ Correspondence theory is the characterization of given

properties of → by means of linguistic schemata, e.g.:

◮ Transitivity is characterized by the schema 4:

✷A ⊃ ✷✷A

◮ Euclideanness: ∀s, t, u : if s → t and s → u then t → u

by the schema 5: ✸A ⊃ ✷✸A

◮ Converse well-foundedness (not a first-order

condition!) by the L¨

• b schema ✷(✷A → A) → ✷A (in

the context of 4).

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

Beyond Basic Modal Logic

◮ The view of modal logic as the theory of state

transitions subsumes other accounts:

• 1. The alethic modality ✷ connects a state s to a state s′

representing a state of affairs that is possible relative to s.

• 2. The deontic modality connects a state s to a state s′

where all s-obligations are fulfilled.

• 3. The epistemic modality K connects a state s to a state

s′ which is consistent with what the agent knows at s.

◮ The account can also be generalized along several

different directions:

• 1. Allowing more than one kind of transition

(poly-modal logic or labeled transition systems);

• 2. Constraining the number of out-going arrows from s

(graded modalities);

• 3. Using binary modalities such as until(p, q).

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

Semantic Networks

◮ In the 1970’s a number of direct (i.e., not logic-based)

approaches were developed for the representation of specialized knowledge bases.

◮ Among these are semantic networks, where nodes

refer to classes of individuals, edges represent IS-A (subsumption) links, as well as, possibly value restrictions.

◮ Such networks support assertions obtained by

chaining through IS-A links, and they provide a simple yet powerful mechanism for knowledge representation.

◮ The problem is that such networks lack a

well-defined semantics.

A semantic network

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

Description Logics

◮ Description Logics, initially known also as

terminological systems, provide a mathematically precise representation for this kind of networks.

◮ Description logics are used to provide “ontologies” for

many different fields, from medicine, to software enginnering, to library science.

◮ The language of DL is built up from concepts C, D, . . .

(1-place preds) and roles (2-place preds) R, S, . . . by means of several operations: C, D → c ⊤ C ⊓ D ¬C ∀R. C

◮ Notice that in this version of DL only atomic roles are

allowed, but we have full negation.

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

Semantics for DL

◮ Given a non-empty, possibly infinite domain U, we

define an interpretation E assigning subsets of U to atomic concepts and subsets of U2 to (atomic) roles. The interpretation can then be lifted as follows:

◮ E [⊤] = U ◮ E [C ⊓ D] = E [C] ∩ E [D] ◮ E [¬C] = U \ E [C] ◮ E [∀R. C] = {d ∈ U : ∀e ∈ U(d, e ∈ E [R] → e ∈ E [C])}

◮ We can then take ∃, ⊔, and ⊥ as defined . . . ◮ . . . or extend the language by number restrictions:

E [≤ nR] = {d : card{e : E [R](d, e)} ≤ n}

◮ and non-atomic, i.e., compound, roles (more about

this later).

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

Examples in DL

◮ The set of all women: Person ⊓ Female ◮ The set of all parents: Person ⊓ ∃HasChild. ⊤ ◮ The set of parents of only daughters:

Person ⊓ ∀HasChild. Female

◮ the set of all childless people: Person ⊓ ∀HasChild. ⊥ ◮ the set of parents of only children:

Person ⊓ ∃HasChild. ⊤⊓ ≤ 1HasChild Notice that all these statements are variable-free.

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

From Description Logic to Modal Logic

◮ It was first noticed by K. Schild (1991) that it is

natural to interpret the domain U as a set of possible worlds and concepts C as propositions, i.e., sets of possible worlds at which the proposition holds.

◮ On this interpretation, the ∀. operator of DL (with

• nly atomic roles) becomes a modal operator and

each atomic role r becomes an accessibility relation.

◮ This way we obtain a translation into Km,

multi-modal K.

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

Axiomatizing DL

◮ Say that D subsumes C, written |

= C ⊑ D, iff E [C] ⊆ E [D] for every interpretation E .

◮ Similarly say that C and D are equivalent, written

| = C = D, iff E [C] = E [D] for every interpretation E .

◮ Subsumption is reducible to equivalence for C ⊑ D iff

C ⊓ D = C

◮ We are interested in axiomatizing the equational

theory of DL (with atomic roles only).

◮ The translation into Km immediately gives the

following axioms:

◮ Axioms forcing (⊤, ⊓, ¬) to be a Boolean Algebra; ◮ ∀R. ⊤ = ⊤ ◮ ∀R.(C ⊓ D) = (∀R. C) ⊓ (∀R. D). ◮ From Km we also obtain that subsumption is (not only

decidable, but) PSPACE-complete.

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

Propositional Dynamic Logic

◮ PDL is based on Vaughan Pratt’s idea to associate

with each non-deterministic program α a distinct modality [α].

◮ PDL takes the idea of multiple modalities very

seriously and applies it to the analysis and representation of computation.

◮ The language of PDL is built up from formulas and

programs, recursively defined: A, B → a ¬A A ∨ B [α]A α, β → p (α ; β) (α ∪ β) α∗ A?

◮ The abbreviations 1, ∧, →, and α are as usual. ◮ Standard programming constructs can be

represented, e.g.:

◮ if A then α else β

as ((A? ; α) ∪ ((¬A? ; β)

◮ while A do α

as ((A? ; α)∗ ; ¬A?)

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

Kripke semantics for PDL

◮ A model structure M for PDL is a triple (W, V, R),

where W is set of worlds, V maps atomic propositions into subsets of W and R maps atomic programs into subsets of W2.

◮ V and R can be lifted to complex formulas and

programs by simultaneous recursion. Here are the clauses for R: R(α ; β)(u, v) ⇐ ⇒ ∃w[R(α)(u, w) & R(β)(w, v) R(α ∪ β)(u, v) ⇐ ⇒ R(α)(u, v) or R(β)(u, v) R(α∗)(u, v) ⇐ ⇒ u, v is in the trasitive closure of R(α) R(A?)(u, v) ⇐ ⇒ u = v & u ∈ V(A)

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

Axiomatizing PDL

◮ The following set of axioms is sound and complete

for PDL:

• 1. [α ; β]A ↔ [α][β]A
• 2. [α ∪ β]A ↔ [α]A ∧ [β]A
• 3. [α∗]A ↔ A ∧ [α][α∗]A
• 4. [A?]B ↔ (A → B)

◮ with the following rule:

A → [α]A A → [α∗]A

◮ The last axiom does not allow the eliminations of

tests from PDL!

◮ Moreover, PDL has the finite model property and is

therefore decidable. Satisfiability for PDL is in fact NEXPTIME-complete.

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

Variants of PDL

◮ There are many variants and extensions for PDL. We

mention two.

◮ PDL with deterministic programs: then R(α) is a

partial function on W.

◮ DPDL is axiomatized by adding to PDL the single

axiom schema: αA → [α]A

◮ PDL with converse: we introduce a converse operator

• n accessibility relations, α−1, where R(α−1)(u, v)

holds iff R(α)(v, u).

◮ An axiomatization is obtained by adding the axioms:

A → [α]α−1A A → [α−1]αA

◮ PDL with converse is not significantly different in

complexity from PDL.

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

Description Logics with Compound Roles

◮ We extend DL to allow for compound roles:

R, S → r R ◦ S R ⊔ S R∗ R−1 id(C) where id(C) is the identity on C, and extend the semantic interpretation function E accordingly.

◮ The converse operator −1 commutes with ⊔, ◦, and ∗

and so it only need be applied to atomic roles.

◮ Also define R+ := R ◦ R∗ and self := id(⊤).

◮ This version of DL is just a notational variant of PDL

with converse — roles are re-interpreted as non-deterministic programs; e.g:

• 1. ∀R. C: C holds whenever R terminates;
• 2. R1 ◦ R2: run R1 and then R2;
• 3. R1 ⊔ R2: non-det’ly run one of R1, R2;
• 4. R∗: non-det’ly pick n ≥ 0 and run R ◦ · · · ◦ R
• n times

.

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

Axiomatizing DL with Compound Roles

◮ The correspondence with converse PDL also gives an

axiomatization:

• 1. ∀(R ⊔ S). C = (∀R. C) ⊓ (∀S. C)
• 2. ∀(R ◦ S). C = ∀R. ∀S. C
• 3. ∀id(C). D = ¬C ⊔ D
• 4. ∀R∗ . C = C ⊓ ∀(R+). C
• 5. C ⊓ ∀R∗ .(¬C ⊔ ∀R. C) ⊑ ∀R∗ . C
• 6. C ⊑ ∀r∃r−1 . C
• 7. C ⊑ ∀r−1∃r. C

◮ Just like converse PDL, DL with compound roles has

the finite model property. Subsumption is therefore decidable, in fact decidable in NEXPTIME.

◮ Note that the finite model property is lost with

intersection of roles: ∀r∗ .((∃r. ⊤) ⊓ ∀(r+ ⊓ self). ⊥ is an axiom of infinity giving an acyclic r-chain.

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

The modal µ-calculus

◮ The modal µ-calculus significantly extends PDL by

introducing a least fixed-point operator µ. The result is strictly more expressive than PDL while still EXPTIME (-complete).

◮ Formulas are built up from propositional variables

and propositional as well as program constants: φ, ψ → x p φ ∨ ψ ¬φ aφ µx. φ(x), the last clause with the proviso that X must occur positively in φ. Notice that only atomic programs are needed.

◮ We can define 1, ∧, → and [a] as usual, as well

greatest fixed-points: νx. φ(x) := ¬µx. ¬φ(¬x)

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

Semantics of the modal µ-calculus

◮ As in PDL, a model M is a triple (W, V, R), with

V(0) = ∅. A (partial) valuation assigns subsets of W to (some of) the variables x, y, z, . . .

◮ If A = A1, . . . , An is a partial valuation (with each

Ai ⊆ W) and φ(x) is a formula (with x = x1, . . . , xn), the extension φM(A) of φ is defined inductively by putting xM

i (A) = Ai and pM(A) = V(p), and

¬φM(A) = W \ φM(A), (φ ∨ ψ)M(A) = φM(A) ∪ ψM(A), aφM(A) = {w ∈ W : ∃v ∈ φM(A). R(a)(v, w)}, µx. φ(x)M(A) = {B ⊆ W : φM(B, A) ⊆ B}

◮ The last line is justified because if x is positive in φ

then φM defines a monotone operator over P(W), and so by the Knaster-Tarski theorem it has a least as well as a greatest fixed point.

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

Expressive power of the modal µ-calculus

◮ PDL is properly subsumed by the modal µ-calculus. ◮ Some fixed points can be represented in PDL, viz.,

those of the form a∗φ.

◮ In fact, a∗φ is the least fixed point of the operator

φ ∨ ax, and is therefore represented by µx. φ ∨ ax

◮ The operator φ ∨ ax is continuous in x (commutes

with ) and it therefore closes at the ordinal ω.

◮ Consider instead

µx.[a]x = {w : there are no infinite a-paths out of w}

◮ The operator [a]x is not continuous: it closes at ω + 1

• n the tree of all sequences n, s, where n > 0 and s

is a finite word over a 1-letter alphabet of length ≤ n, with as a top node.

◮ In fact µx.[a]x is not equivalent to any PDL-formula.

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

Second-order propositional bi-modal logic

◮ The language of 2S5 is obtained from propositional

variables, connectives, two distinct S5-modalities, ✷

1

and ✷

2 , and propositional quantifiers ∀p.

◮ A model M is, as before, a tuple (W, R1, R2, V), where

each Ri is an equivalence over W, and V assigns subsets of W to the propositional variables.

◮ Propositional quantifiers are given the standard

interpretation, in that they range over P(W).

◮ M, w |

= ∀pφ iff M[X/p] | = φ for every X ⊆ W, where M[X/p] assigns X to p and is otherwise like M.

◮ In 2S5 we can write, for instance:

∀p✷

1 (p → ∃q✸ 2 q)

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

The complexity of 2S5

◮ The mono-modal case: mono-modal S5 with

propositional quantifiers can be interpreted into monadic second-order logic and is therefore decidable (as proved independently by D. Kaplan and K. Fine in 1970).

◮ Asymmetric modal logics: each of the modal logics K,

B, T, K4, S4, when augmented with propositional quantifiers (with the standard interpretation) becomes equivalent to full second-order logic. The asymmetric nature of the accessibility relation allows to define a notion of order that can be used to represent second-order logic.

◮ The bi-modal case: The set of validities 2S5 is

effectively equivalent to full second-order logic (under the standard interpretation), and hence not axiomatizable (Antonelli & Thomason, 2001). In this respect, the decidability of second-order S5 is an anomaly, and the poly-modal case is the more natural

• ne.

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

The epistemic interpretation

◮ S5 is a natural first candidate for epistemic

modalities.

◮ S5 conveys that knowledge implies truth, as well as

both positive (4) and negative (5) introspection: ✷φ → φ ✷φ → ✷✷φ ¬✷φ → ✷¬✷φ

◮ In the case of multiple agents, each reasoning about

the others’ as well as his or her own knowledge, one needs multiple modalities.

◮ In fact, certain reasoning tasks require common

knowledge (puzzle of the muddy children).

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

Common Knowledge

◮ Agents i and j have common knowledge (CK) of a

proposition p iff the following proposition q is true: p, and both i and j know that q.

◮ The inherent circularity can be analyzed iteratively as

the infinite conjunction: p∧✷

i p∧✷ j p∧✷ i ✷ i p∧✷ j ✷ j p∧✷ i ✷ j p∧✷ j ✷ i p∧. . .

◮ Agents i and j have common knowledge that p (at a

world w) iff p is true at all worlds v that can be reached from w by a finite sequence of i- and j-links.

◮ Thus common knowledge requires the transitive

closure of the Ri relations. Such transitive closure is explicitly definable in 2S5, which provides a natural framework for the formalization of this kind of reasoning tasks.

Philosophical Logic and Computer Science Aldo Antonelli Philosophical Logic Knowledge Representation Forays into the multi-modal world Going Second-Order Propositional Conclusion

Conclusion

◮ We have come full circle: we started out with some

quintessentially philosophical issues (the nature of possibility and necessity), then moved on to the analysis

• f transition systems of many different kinds.

◮ This has led us to re-evaluate the significance of the

poly-modal framework and how distinctly different it can be from the mono-modal case.

◮ In turn, this leads to further extending the propositional

language at the second-order, in search of more and more expressive resources.

◮ And finally we come to the application of the resulting

logical framework to the analysis of philosophical problems, such as the nature of reflective knowledge.

◮ Thus we should re-evaluate well-entrenched distinctions

between philosophical, mathematical, computational logic: It’s all just (symbolic) logic.